Probability mass function

The probability mass function assigns a probability P(X = x) to each real number x ∈ ℝ:

The probability mass function f: ℝ ⟶ [0,1] of a discrete random variable X with support \(T = {x_1, x_2 ... , x_k, ... }\) is defined as
\[f(x) = \begin{cases} P(X = x_i) = p_i ,& x = x_i \in {x_1,...x_k,...} \\ 0 ,& otherwise \end{cases} \]

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For the binomial distribution

A (discrete) random variable X is called binomially distibuted with parameters n and \(\pi\) (for short: B(n, \(\pi\))-distributed or ***X ~ B(n, \(\pi\)), if it has the following probability mass function:

\[f(x) = \begin{cases} \binom{n}{x}\pi^x(1-\pi)^{n-x} ,& x = 0, 1, \dots, n\\ 0 ,& otherwise \end{cases} \]

For the Poisson distribution

\[f(x) = \begin{cases} \frac{\lambda^x}{x!} e^{-\lambda} ,& x \in {0, 1, \dots}\\ 0 ,& otherwise \end{cases} \]

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